Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-14T17:31:51.014Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic periodicity in outer billiards with contraction

Published online by Cambridge University Press:  14 June 2018

JOSÉ PEDRO GAIVÃO
JOSÉ PEDRO GAIVÃO*
Affiliation:
Departamento de Matemática e CEMAPRE, ISEG, Universidade de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa, Portugal email jpgaivao@iseg.utl.pt
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that for almost every $(P,\unicode[STIX]{x1D706})$, where $P$ is a convex polygon and $\unicode[STIX]{x1D706}\in (0,1)$, the corresponding outer billiard about $P$ with contraction $\unicode[STIX]{x1D706}$ is asymptotically periodic, i.e., has a finite number of periodic orbits and every orbit is attracted to one of them.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 2 , February 2020 , pp. 402 - 417
Copyright
© Cambridge University Press, 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adler, R., Kitchens, B. and Tresser, C.. Dynamics of non-ergodic piecewise affine maps of the torus. Ergod. Th. & Dynam. Sys. 21(4) (2001), 959–999.Google Scholar
Ashwin, P. and Goetz, A.. Polygonal invariant curves for a planar piecewise isometry. Trans. Amer. Math. Soc. 358(1) (2006), 373–390.Google Scholar
Beaucoup, F., Borwein, P., Boyd, D. W. and Pinner, C.. Multiple roots of [-1, 1] power series. J. Lond. Math. Soc. (2) 57(1) (1998), 135–147.Google Scholar
Borwein, P. and Erdélyi, T.. Polynomials and Polynomial Inequalities (Graduate Texts in Mathematics, 161) . Springer, New York, 1995.Google Scholar
Brémont, J.. Dynamics of injective quasi-contractions. Ergod. Th. & Dynam. Sys. 26(1) (2006), 19–44.Google Scholar
Bruin, H. and Deane, J. H. B.. Piecewise contractions are asymptotically periodic. Proc. Amer. Math. Soc. 137(4) (2009), 1389–1395.Google Scholar
Buzzi, J.. Piecewise isometries have zero topological entropy. Ergod. Th. & Dynam. Sys. 21(5) (2001), 1371–1377.Google Scholar
Catsigeras, E., Guiraud, P., Meyroneinc, A. and Ugalde, E.. On the asymptotic properties of piecewise contracting maps. Dyn. Syst. 31(2) (2016), 107–135.Google Scholar
Day, M. M.. Polygons circumscribed about closed convex curves. Trans. Amer. Math. Soc. 62 (1947), 315–319.Google Scholar
Del Magno, G., Gaivão, J. P. and Gutkin, E.. Dissipative outer billiards: a case study. Dyn. Syst. 30(1) (2015), 45–69.Google Scholar
Eliasson, L. H.. Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum. Acta Math. 179(2) (1997), 153–196.Google Scholar
Goetz, A.. Dynamics of piecewise isometries. Illinois J. Math. 44(3) (2000), 465–478.Google Scholar
Gutkin, E.. Billiard dynamics: an updated survey with the emphasis on open problems. Chaos 22(2) (2012),026116, 13 pp.Google Scholar
Gutkin, E. and Simányi, N.. Dual polygonal billiards and necklace dynamics. Comm. Math. Phys. 143(3) (1992), 431–449.Google Scholar
Haller, H.. Rectangle exchange transformations. Monatsh. Math. 91(3) (1981), 215–232.Google Scholar
Hooper, W. P.. Renormalization of polygon exchange maps arising from corner percolation. Invent. Math. 191(2) (2013), 255–320.Google Scholar
Jeong, I.-J.. Outer billiards with contraction. Senior Thesis, Brown University, 2012.Google Scholar
Jeong, I.-J.. Outer billiards with contraction: attracting Cantor sets. Exp. Math. 24(1) (2015), 53–64.Google Scholar
Jeong, I.-J.. Outer billiards with contraction: regular polygons. Dyn. Syst. doi:10.1080/14689367.2017.1402295. Published online 30 November 2017.Google Scholar
Klein, S.. Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a Gevrey-class function. J. Funct. Anal. 218(2) (2005), 255–292.Google Scholar
Kołodziej, R.. The antibilliard outside a polygon. Bull. Pol. Acad. Sci. Math. 37(1–6) (1989), 163–168.Google Scholar
Kruglikov, B. and Rypdal, M.. Entropy via multiplicity. Discrete Contin. Dyn. Syst. 16(2) (2006), 395–410.Google Scholar
Moser, J.. Is the solar system stable? Math. Intelligencer 1(2) (1978–1979), 65–71.Google Scholar
Neumann, B. H.. Sharing ham and eggs. Iota, the Manchester University Mathematics Students Journal (1) (1958), 14–18.Google Scholar
Nogueira, A. and Pires, B.. Dynamics of piecewise contractions of the interval. Ergod. Th. & Dynam. Sys. 35(7) (2015), 2198–2215.Google Scholar
Nogueira, A., Pires, B. and Rosales, R. A.. Asymptotically periodic piecewise contractions of the interval. Nonlinearity 27(7) (2014), 1603–1610.Google Scholar
Schwartz, R. E.. Unbounded orbits for outer billiards. I. J. Mod. Dyn. 1(3) (2007), 371–424.Google Scholar
Schwartz, R. E.. Outer Billiards on Kites (Annals of Mathematics Studies, 171) . Princeton University Press, Princeton, NJ, 2009.Google Scholar
Schwartz, R. E.. The Octogonal PETs (Mathematical Surveys and Monographs, 197) . American Mathematical Society, Providence, RI, 2014.Google Scholar
Solomyak, B.. On the random series ∑ ±𝜆 n (an Erdös problem). Ann. of Math. (2) 142(3) (1995), 611–625.Google Scholar
Tabachnikov, S.. Geometry and Billiards (Student Mathematical Library, 30) . American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005.Google Scholar
Vivaldi, F. and Shaidenko, A. V.. Global stability of a class of discontinuous dual billiards. Comm. Math. Phys. 110(4) (1987), 625–640.Google Scholar